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Abstract
We give a necessary and sufficient condition for the following property of an integer d∈N and a pair (a,A)∈R^{2}: There exist κ>0 and Q0∈N such that for all x∈Rd and Q≥Q_{0}, there exists p/q∈Qd such that 1≤q≤Q and xp/q≤κq^{a}Q^{A}. This generalizes Dirichlet's theorem, which states that this property holds (with κ=Q_{0}=1) when a=1 and A=1/d. We also analyze the set of exceptions in those cases where the statement does not hold, showing that they form a comeager set. This is also true if Rd is replaced by an appropriate "Diophantine space", such as a nonsingular rational quadratic hypersurface which contains rational points. Finally, in the case d=1 we describe the set of exceptions in terms of classical Diophantine conditions.
Original language  English 

Pages (fromto)  1122 
Number of pages  12 
Journal  Journal of Number Theory 
Volume  162 
Early online date  8 Dec 2015 
DOIs  
Publication status  Published  1 May 2016 
Bibliographical note
© 2015 Elsevier Inc. This is an authorproduced version of the published paper. Uploaded in accordance with the publisher’s selfarchiving policy.Keywords
 Diophantine approximation
 Dirichlet's theorem
Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research